Do calcium channel blockers applied to cardiomyocytes cause increased channel expression resulting in reduced efficacy?

In the initial hours following the application of the calcium channel blocker (CCB) nifedipine to microtissues consisting of human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs), we observe notable variations in the drug’s efficacy. Here, we investigate the possibility that these temporal changes in CCB effects are associated with adaptations in the expression of calcium ion channels in cardiomyocyte membranes. To explore this, we employ a recently developed mathematical model that delineates the regulation of calcium ion channel expression by intracellular calcium concentrations. According to the model, a decline in intracellular calcium levels below a certain target level triggers an upregulation of calcium ion channels. Such an upregulation, if instigated by a CCB, would then counteract the drug’s inhibitory effect on calcium currents. We assess this hypothesis using time-dependent measurements of hiPSC-CMs dynamics and by refining an existing mathematical model of myocyte action potentials incorporating the dynamic nature of the number of calcium ion channels. The revised model forecasts that the CCB-induced reduction in intracellular calcium concentrations leads to a subsequent increase in calcium ion channel expression, thereby attenuating the drug’s overall efficacy. The data and fit models suggest that dynamic changes in cardiac cells in the presence of CCBs may be explainable by induced changes in protein expression, and that this may lead to challenges in understanding calcium based drug effects on the heart unless timings of applications are carefully considered.

In [1,2], the protein regulation model is given in the form Here, c is the cytosolic calcium concentration, c * is the target calcium concentration, g is the conductance of the considered type of ion channel, r is a intermediary variable, and τ r and τ g are time constants.
In our computations, we have rewritten the original formulation (1)-(2) to instead of r and g, model unitless scaling factors n and m for the number of ion channels in the cell membrane and the number of messenger RNAs (mRNAs).More specifically, the number of ion channels and the number of mRNAs are given by N (t) = n(t)N 0 and M (t) = m(t)M 0 , respectively, where N 0 and M 0 are the default number of channels and mRNAs.
The total current through all the I CaL channels in the cell membrane can be expressed as where g 0 is the single channel conductance of the channel, o is the open probability of the calcium channels and i CaL,0 is an expression for how the single-channel current depends on model variables like the membrane potential and the cytosolic calcium concentration.In this setting, the conductance g in the model ( 1)-( 2) is given by where ḡ = N 0 • g 0 is the default value of g (corresponding to n = 1).In other words, we have n = g ḡ ( 5) In order to rewrite (1)-(2) to a system for n and m, we similarly define and divide both ( 1) and ( 2) by ḡ.This yields which can be rewritten as where

Supplementary Note 3: Comparison of nifedipine block percentages from literature
In order to compare the assumed block percentages associated with 0.1 µM and 1 µM of nifedipine, we identified dose-dependent effects on I CaL from literature.The selected block percentages are compared to the data from literature in Supplementary Figure 2. The figure shows the value of b(D) as a function of the dose, D, reported in different studies in addition to the values used in this study.More specifically, the effect of the drug is typically reported in terms on a half-maximal effective concentration, EC 50 , a Hill coefficient, h and a maximal effect E, and b(D) is assumed to be given by Here, as an example, E = −1 if the maximal effect of the drug is to block the current completely, and E = −0.[10] 0.038 µM Supplementary Table 1: Dose-dependent effects of nifedipine on the L-type calcium current from literature used in Supplementary Figure 2. The blocking effect is modeled using (13).In Supplementary Figures 3 and 4, we report the experimental voltage traces used to compute the biomarkers reported in Figure 4 in the paper.We show the baseline traces as well as the traces recorded 2 h, 4 h, 6 h, 8 h, 13 h and 16 h after application of nifedipine.For the dose of 0.1 µM of nifedipine, we have data from five different tissues of hiPSC-CMs, and for 1 µM, we have data from seven tissues.The traces are recorded using voltage sensitive dyes, and we consider the average fluorescence (spatially) over each tissue.The APD50 and APD80 values reported in Figure 4 in the paper are computed using the representative single AP traces marked in orange in the plots and the beat rate is computed using the average of all the recorded beats.The representative single AP traces are selected as the AP with the smallest total difference to the median (temporally) of the biomarkers computed for each of the APs in the trace.In Figure 4 in the paper, the APD and beat rate values are represented by a single dot for each of the considered tissues.Supplementary Figure 3: Experimental voltage traces recorded using voltage sensitive dyes for baseline and after 2 h, 4 h, 6 h, 8 h, 13 h, and 16 h application of 0.1 µM of nifedipine.The APD50 and APD80 values reported in Figure 4 in the paper are computed using the representative traces marked in orange and the beat rate is computed from all the recorded beats.Supplementary Figure 4: Experimental voltage traces recorded using voltage sensitive dyes for baseline and after 2 h, 4 h, 6 h, 8 h, 13 h, and 16 h application of 1 µM of nifedipine.The APD50 and APD80 values reported in Figure 4 in the paper are computed using the representative traces marked in orange and the beat rate is computed from all the recorded beats.
Supplementary Note 5: Investigating the effect of adjusting the number of different protein types in other membrane models In Supplementary Figure 5, we investigate the effect of increasing the number of different types of proteins in three other membrane models.More specifically, we consider the hiPSC-CM models of Paci et al. 2013 [3] and Kernik et al. 2019 [4].In addition, we consider the sinoatrial node (SAN) model of Severi et al. 2012 [5], which is the model used in [2].We observe that for all the considered models, the intracellular calcium concentration increases when the number of some of the protein types increases and that the intracellular calcium concentration increases decreases when the number of other protein types increases.

(SAN)
Supplementary Figure 5: Investigation of the effect on the cytosolic calcium concentration of adjusting the number of different types of proteins (channels, pumps and exchangers) in the other models.Ase report the percent change in the average cytosolic calcium concentration resulting from a 20% increase in the number of each of the types of channels, pumps and exchangers in the model.The effects are measured by comparing the average cytosolic calcium concentration over 10 seconds of simulation, 2 minutes after the parameter change was applied, and comparing it to the average (over 10 seconds) in a simulation of the default model.
In this note, the formulation of the base model for the action potential of hiPSC-CMs, based on [11,12] is provided.In the model formulation, the membrane potential (v) is given in units of mV, the Ca 2+ concentrations are given in units of mM, all currents are expressed in units of A/F, and the Ca 2+ fluxes are expressed as mmol/ms per total cell volume (i.e., in units of mM/ms).The parameters of the model are given in Supplementary Tables 2-8.

Membrane potential
The membrane potential is governed by the equation where I stim is an applied stimulus current, and I Na , I NaL , I CaL , I to , I Kr , I Ks , I K1 , I NaCa , I NaK , I pCa , I bCl , I bCa , and I f are membrane currents specified below.

Membrane currents
The currents through the voltage-gated ion channels on the cell membrane are in general given on the form where g is the channel conductance, v is the membrane potential and E is the equilibrium potential of the channel.Furthermore, o = i z i is the open probability of the channels, where z i are gating variables, either given as a function of the membrane potential or governed by equations of the form The parameters τ z i and z i,∞ are specified for each of the gating variables of the model in Supplementary Table 9.
Fast sodium current The formulation of the fast sodium current is an adjusted version of the model given in [13], supporting slower upstroke velocities more similar to those observed in the optical measurements of hiPSC-CMs.The current is given by where the open probability is given by and m and j are gating variables governed by equations of the form (15).

Late sodium current
The formulation of the late sodium current, I NaL , is based on [14] and is given by where the open probability is given by and m L and h L are gating variables governed by equations of the form (15).

Transient outward potassium current
The formulation of the transient outward potassium current, I to , is based on [3] and is given by where the open probability is given by and q to and r to are gating variables governed by equations of the form (15).

Rapidly activating potassium current
The formulation of the rapidly activating potassium current, I Kr , is based on [3] and is given by where and the dynamics of x Kr1 and x Kr2 are governed by equations of the form (15).

Slowly activating potassium current
The formulation of the slowly activating potassium current, I Ks , is based on [13] and is given by where and the dynamics of x Ks is governed by an equation of the form (15).

Inward rectifier potassium current
The formulation of the inward rectifier potassium current, I K1 , is based on [13] and is given by where Hyperpolarization activated funny current The formulation for the hyperpolarization activated funny current, I f , is based on [3] and is given by where and the dynamics of x f is governed by an equation of the form (15).
L-type Ca 2+ current The formualtion for the L-type Ca 2+ current, I CaL , is based on the formulation in [13] and is given by where and the dynamics of d, f and f Ca are governed by equations of the form (15). Furthermore, n is the scaling factor for the number of L-type calcium channels in the cell membrane.In the paper, we consider two alternative models for this factor, one model directly based on [1,2] given by ( 9)-( 10) and one updated model described in the main paper and repeated in (69)-(71) below.

Background currents
The formulation of the background currents, I bCa and I bCl , are based on [13] and are given by Sodium-calcium exchanger The formulation of the Na + -Ca 2+ exchanger current, I NaCa , is based on [13] and is given by where Sarcolemmal Ca 2+ pump The formulation of the current through the sarcolemmal Ca 2+ pump, I pCa , is based on [13] and is given by Sodium-potassium pump The current through the Na + -K + pump, I NaK , is based on [13] and is given by where

Nernst equilibrium potentials
The Nernst equilibrium potentials for the ion channels are defined as for the parameter values given in Supplementary Table 3.

Ca 2+ dynamics
The Ca 2+ dynamics are governed by where c d is the concentration of free Ca 2+ in the dyad, b d is the concentration of Ca 2+ bound to a buffer in the dyad, c sl is the concentration of free Ca 2+ in the SL compartment, b sl is the concentration of Ca 2+ bound to a buffer in the SL compartment, c c is the concentration of free Ca 2+ in the bulk cytosol, b c is the concentration of Ca 2+ bound to a buffer that is not troponin in the bulk cytosol, b t is the concentration of Ca 2+ bound to troponin in the bulk cytosol, c s is the concentration of free Ca 2+ in the jSR, b s is the concentration Buffer fluxes The fluxes of free Ca 2+ binding to a Ca 2+ buffer are given by Membrane fluxes The membrane fluxes, J CaL , J bCa , J pCa , and J NaCa , are given by where I CaL , I bCa , I pCa , and I NaCa are defined by the expressions given above.Furthermore, J sl e = J NaCa + J pCa + J bCa .
Na + dynamics For the intracellular Na + concentration, we use the same approach as in [12].In this approach, spatial gradients of [Na + ] i in the cell are ignored and the concentration is governed by where the currents I Na , I NaL , I NaK , I NaCa , and I f are specified above.

Protein regulation
The scaling factor n for the number of L-type calcium channels on the cell membrane is modeled by   α xKs Supplementary Table 9: Specification of the parameters z ∞ and τ z , for z = m, j, m L , h L , d, f, f Ca , q to , r to , x Kr1 , x Kr2 , x Ks and x f in the equations for the gating variables (15).

Supplementary Figure 1 :
Time evolution of the solutions n and m of the model (9)-(10) coupled to an action potential model of hiPSC-CMs for different values of τ m .The simulations are started from n = m = 0.1, and we use τ m = 400 mMms.

Supplementary Figure 2 :
Comparison of the blocking effect of nifedipine b(D) used in our model to data from literature.

Table 8 :
Parameters for the regulation of the number of Ltype calcium channels in the cell membrane.